Suppose that f = u + iv is analytic in a domain D. Show that the Cauchy-Riemann equations in polar coordinates (r, 0), where x =r cos 0, y = r sin 0, are given by du dv ar ne ||

Use the polar form of C-R equations to verify that C-R conditions hold in case of f:C +C where f(2) = z12, (z e C). Then verify that f'(a) = 12a" for all points а.

2.Make a sketch of the following polar equation: r = 2 +4 sin 0 (r, 0). Determine also the tangent gradient of the polar curve at the point with 0 = "/2

The slope of the line tangent to the polar graph r = 0 at the point where 0 = 1 is: